Gauging non-Hermitian Hamiltonians

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that a limited kind of gauge transformation in X implies minimal substitution in the form P → P − eA. We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian h is local, and in the more generic example of the imaginary cubic interaction, where H is local but h is not.